Devise a formula to reveal the uncertainty on x∗ due to uncertainty on the “givens.” You choose a model. I do this by looking at g, the derivative of the OF w.r.t. x. At the optimum g = 0. At f...

Devise a formula to reveal the uncertainty on x∗ due to uncertainty on the “givens.” You choose a model. I do this by looking at g, the derivative of the OF w.r.t. x. At the optimum g = 0. At f perturbed by a deviation in a given, g will be shifted a bit. If the changes are small, you can estimate the amount of x∗ shift from the g graph. For propagation of maximum uncertainty, I get

Here, ci represents one of the model coefficients. This part of the exercise is a mathematical analysis. Note the derivatives are the commonly occurring Jacobean (gradient) and Hessian elements with the additional c partial derivative. In any practical application, the derivatives would have to be evaluated numerically. Also note that if there are several uncertain values, it is unlikely that each will be at its extreme value and that each will push the x∗ in the same direction. Propagation of probable error provides a more plausible value than propagation of maximum error.